Degree estimate for commutators

Let K be a free associative algebra over a field K of characteristic 0 and let each of the noncommuting polynomials f,g generate its centralizer in K. Assume that the leading homogeneous components of f and g are algebraically dependent with degrees which do not divide each other. We give a counterexample to the recent conjecture of Jie-Tai Yu that deg([f,g])=deg(fg-gf) > min{deg(f),deg(g)}. Our example satisfies deg(g)/2 < deg([f,g]) < deg(g) < deg(f) and deg([f,g]) can be made as close to deg(g)/2 as we want. We obtain also a counterexample to another related conjecture of Makar-Limanov and Jie-Tai Yu stated in terms of Malcev - Neumann formal power series. These counterexamples are found using the description of the free algebra K considered as a bimodule of K[u] where u is a monomial which is not a power of another monomial and then solving the equation [u^m,s]=[u^n,r] with unknowns r,s in K.Comment: 18 page

Modified bubble level senses pitch and roll angles over wide range

Bubble level sensor with fiber-optic field flattener is simple, rugged, small, and impervious to temperature and vibration effects. Pitch angles from -15 deg to +40 deg and roll angles of +30 deg are determined within 0.5 deg

A note on the arithmetic properties of Stern Polynomials

We investigate the Stern polynomials defined by $B_0 ( t ) =0,B_1 ( t ) =1$, and for $n \geq 2$ by the recurrence relations $B_{2n}( t) =tB_{n}( t) ,$ $B_{2n+1}( t) =B_n( t) +B_{n+1}( t)$. We prove that all possible rational roots of that polynomials are $0,-1,-1/2,-1/3$. We give complete characterization of $n$ such that $deg( B_n) = deg( B_{n+1})$ and $deg( B_n) =deg( B_{n+1}) =deg( B_{n+2})$. Moreover, we present some result concerning reciprocal Stern polynomials.Comment: 9 pages, submitte

Comet Machholz (C/2004 Q2): morphological structures in the inner coma and rotation parameters

Extensive observations of comet C/2004 Q2 (Machholz) were carried out between August 2004 and May 2005. The images obtained were used to investigate the comet's inner coma features at resolutions between 350 and 1500 km/pixel. A photometric analysis of the dust outflowing from the comet's nucleus and the study of the motion of the morphological structures in the inner coma indicated that the rotation period of the nucleus was most likely around 0.74 days. A thorough investigation of the inner coma morphology allowed us to observe two main active sources on the comet's nucleus, at a latitude of +85{\deg} \pm 5{\deg} and +45{\deg} \pm 5{\deg}, respectively. Further sources have been observed, but their activity ran out quite rapidly over time; the most relevant was at latcom. = 25{\deg} \pm 5{\deg}. Graphic simulations of the geometrical conditions of observation of the inner coma were compared with the images and used to determine a pole orientation at RA=95{\deg} \pm 5{\deg}, Dec=+35{\deg} \pm 5{\deg}. The comet's spin axis was lying nearly on the plane of the sky during the first decade of December 2004.Comment: 29 pages, 8 figures, 3 table